Spinors as automorphisms of the tangent bundle

2002/10/31 by Alexandru Scorpan
Mathematics · #math.DG #math.GT #msc:53C27 #msc:57N13 #msc:32Q60 #msc:53D05

paper · pdf

published as Trans. Amer. Math. Soc., vol. 356 (2004), mo. 5, pp. 2049-2066 · 19 pages, 1 LaTeX figure. Minor revision, one figure added. To appear in Transaction of the AMS

arxiv created 2003/04/11 · arxiv updated 2009/11/30

Abstract

We show that, on a 4-manifold M endowed with a spin^c structure induced by an almost-complex structure, a self-dual (= positive) spinor field ϕ∈ Γ(W⁺) is the same as a bundle morphism ϕ: TM → TM acting on the fiber by self-dual conformal transformations, such that the Clifford multiplication is just the evaluation of ϕon tangent vectors, and that the squaring map σ: W⁺ → Λ⁺ acts by pulling-back the fundamental form of the almost-complex structure. We use this to detect Kahler and symplectic structures.

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